Statistical Physics of Run-and-Tumble bacteria Julien Tailleur M. - PowerPoint PPT Presentation

Statistical Physics of Run-and-Tumble bacteria Julien Tailleur M. Cates, D. Marenduzzo, I. Pagonabarraga, A. Thompson Laboratoire MSC CNRS - Université Paris Diderot Large Fluctuations in Non-Equilibrium Systems J. Tailleur (CNRS-Univ Paris Diderot) LAFNES 2011 1 / 27

The big picture Micrometer scale Centimeter scale J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 2 / 27

Outline Intro Effective temperature and non-equilibrium dynamics Pattern formation in bacterial colonies Large deviations of bacteria on lattice J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 3 / 27

Run-and-Tumble bacteria Escherichia coli – Unicellular Organism ( 1 µm × 3 µm ) Flagella (few µm long) Electron microscopy J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 4 / 27

Schematic trajectory [Berg & Brown, Nature, 1972] Run: straight line (velocity v ≃ 20 µ m . s − 1 ) • Tumble: change of direction (rate α ≃ 1 s − 1 , duration τ ≃ 0 . 1 s ) • J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 5 / 27

Schematic trajectory [Berg & Brown, Nature, 1972] Run: straight line (velocity v ≃ 20 µ m . s − 1 ) • Tumble: change of direction (rate α ≃ 1 s − 1 , duration τ ≃ 0 . 1 s ) • J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 5 / 27

Schematic trajectory [Berg & Brown, Nature, 1972] Run: straight line (velocity v ≃ 20 µ m . s − 1 ) • Tumble: change of direction (rate α ≃ 1 s − 1 , duration τ ≃ 0 . 1 s ) • Diffusion at large scale v 2 dα (1+ ατ ) ∼ 100 µm 2 .s − 1 Run-and-Tumble D = 6 πηr ∼ 0 . 2 µm 2 .s − 1 kT Brownian Motion D col = J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 5 / 27

In and out of “equilibrium” 10 µm colloids in a bacterial bath [Wu Libchaber 2000] J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 6 / 27

In and out of “equilibrium” 10 µm colloids in a bacterial bath [Wu Libchaber 2000] t ≫ α − 1 “Effective temperature” t ≪ α − 1 Superdiffusive regime J. Tailleur (CNRS-Univ Paris Diderot) Introduction LAFNES 2011 6 / 27

Master Equation ( d > 1 ) ˙ P ( x, u ) = J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 7 / 27

Master Equation ( d > 1 ) P ( x, u ) = − ∇ · [ P ( x, u ) v ] − α ( u ) P ( x, u ) + 1 � ˙ d u ′ α ( u ′ ) P ( x, u ′ ) Ω J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 7 / 27

Master Equation ( d > 1 ) P ( x, u ) = − ∇ · [ P ( x, u ) v ] − α ( u ) P ( x, u ) + 1 � ˙ d u ′ α ( u ′ ) P ( x, u ′ ) Ω � ρ ( x ) = d u P ( x, u ) J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 7 / 27

True equilibrium... perturbatively... D 0 = v 2 If v ( u ) = v u then ρ = ˙ D 0 ∆ ρ with dα t ≫ 1 α J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 8 / 27

True equilibrium... perturbatively... D 0 = v 2 If v ( u ) = v u then ρ = ˙ D 0 ∆ ρ with dα t ≫ 1 α Sedimentation v ( u ) = v u + v τ v τ = − µ δm g u z with J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 8 / 27

True equilibrium... perturbatively... D 0 = v 2 If v ( u ) = v u then ρ = ˙ D 0 ∆ ρ with dα t ≫ 1 α Sedimentation v ( u ) = v u + v τ v τ = − µ δm g u z with � λ ( v T + v ) + α � 2 λv ρ ( z ≫ v/α ) ∝ e − λz ; = ln α λ ( v T − v ) + α J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 8 / 27

True equilibrium... perturbatively... D 0 = v 2 If v ( u ) = v u then ρ = ˙ D 0 ∆ ρ with dα t ≫ 1 α Sedimentation v ( u ) = v u + v τ v τ = − µ δm g u z with � λ ( v T + v ) + α � 2 λv ρ ( z ≫ v/α ) ∝ e − λz ; = ln α λ ( v T − v ) + α λ = v τ If v τ /v ≪ 1 D 0 ρ ∝ e − µ δm g z/D 0 J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 8 / 27

True equilibrium... perturbatively... D 0 = v 2 If v ( u ) = v u then ρ = ˙ D 0 ∆ ρ with dα t ≫ 1 α Sedimentation v ( u ) = v u + v τ v τ = − µ δm g u z with � λ ( v T + v ) + α � 2 λv ρ ( z ≫ v/α ) ∝ e − λz ; = ln α λ ( v T − v ) + α λ = v τ If v τ /v ≪ 1 D 0 ρ ∝ e − µ δm g z/D 0 = e − β eff V ext ( z ) Effective kT eff = D 0 bacteria ≡ hot colloids µ J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 8 / 27

Generic Potential v ( u ) = v u + v τ where v τ = − µ ∇ V ext J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 9 / 27

Generic Potential v ( u ) = v u + v τ where v τ = − µ ∇ V ext If µ ∇ V ext ≪ v and µ ∇ 2 V ext ≪ α � − µV ext ( � r ) � ρ ( � r ) ∝ exp D 0 J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 9 / 27

Generic Potential v ( u ) = v u + v τ where v τ = − µ ∇ V ext If µ ∇ V ext ≪ v and µ ∇ 2 V ext ≪ α � − µV ext ( � r ) � ρ ( � r ) ∝ exp D 0 Effective Temp. kT eff = D 0 bacteria ≡ Hot colloids µ Non-perturbatively? Only quantitative difference? J. Tailleur (CNRS-Univ Paris Diderot) External Potentials LAFNES 2011 9 / 27

Fishing lobsters at the micrometer scale [ P . Galajda, J. Keymer, P . Chaikin, R. Austin, J. Bacteriol. 189 , 8704 (2007)] J. Tailleur (CNRS-Univ Paris Diderot) A bacterial ratchet LAFNES 2011 10 / 27

Fishing lobsters at the micrometer scale Colloids J. Tailleur (CNRS-Univ Paris Diderot) A bacterial ratchet LAFNES 2011 10 / 27

Fishing lobsters at the micrometer scale Bacteria J. Tailleur (CNRS-Univ Paris Diderot) A bacterial ratchet LAFNES 2011 10 / 27

7 6 5 Ratio 4 3 2 1 0 0 10 20 30 40 50 60 v Where does the ratchet effect come from ? Bacteria align with walls upon collisions •• Asymmetric walls no left-right symmetry Interactions with walls no time-reversal symmetry J. Tailleur (CNRS-Univ Paris Diderot) A bacterial ratchet LAFNES 2011 11 / 27

Where does the ratchet effect come from ? Bacteria align with walls upon collisions •• Asymmetric walls no left-right symmetry Interactions with walls no time-reversal symmetry Elastic collisions No ratchet effect 7 6 5 Ratio ρ 2 4 3 ρ 1 2 1 0 0 10 20 30 40 50 60 v v J. Tailleur (CNRS-Univ Paris Diderot) A bacterial ratchet LAFNES 2011 11 / 27

Patterning: Math. Biol. vs Stat. Mech. Specific vs generic (universal?) Quantitative vs qualitative Accurate description vs general mechanisms J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 12 / 27

Pattern formation: extra ingredients [Woodward et al., 1995] Cells division & ‘death’ Interactions (chemical, steric...) clusters are non-motile Is this enough ? J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 13 / 27

Interactions: fluctuating hydrodynamics Microscopic: α , v , τ Mesoscopic: density field ρ ( x, t ) J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 14 / 27

Interactions: fluctuating hydrodynamics Microscopic: α , v , τ Mesoscopic: density field ρ ( x, t ) ρ ( x, t ) = −∇ J ( x, t ); ˙ J ( x, t ) = − D ∇ ρ + V ρ v 2 V = − v v D = dα ∇ dα (1 + ατ ) 1 + ατ J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 14 / 27

Interactions: fluctuating hydrodynamics Microscopic: α , v , τ Mesoscopic: density field ρ ( x, t ) ρ ( x, t ) = −∇ J ( x, t ); ˙ J ( x, t ) = − D ∇ ρ + V ρ v 2 V = − v v 1 + ατ = − D ′ ( ρ ) ∇ ρ/ 2 D = dα ∇ dα (1 + ατ ) Interactions: tumble rate α constant, tumble duration τ = 0 swimming speed v ′ [ ρ ( x )] < 0 D eff ( ρ ) = D ( ρ ) + ρD ′ ( ρ ) / 2 J ( x, t ) = − D eff ( ρ ) ∇ ρ J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 14 / 27

Interactions and phase separation D ( ρ ) = v 2 ( ρ ) /dα D eff ( ρ ) = D ( ρ ) + ρD ′ ( ρ ) / 2 v ′ ( ρ ) < 0 D ′ ( ρ ) < 0 J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 15 / 27

Interactions and phase separation D ( ρ ) = v 2 ( ρ ) /dα D eff ( ρ ) = D ( ρ ) + ρD ′ ( ρ ) / 2 v ′ ( ρ ) < 0 D ′ ( ρ ) < 0 Flat profile ρ ( x ) = ρ 0 unstable if D eff ( ρ 0 ) < 0 Example: v ( ρ ) = v 0 exp( − ρ/ ¯ ρ ) Instability for ρ > ¯ ρ J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 15 / 27

Interactions and phase separation D ( ρ ) = v 2 ( ρ ) /dα D eff ( ρ ) = D ( ρ ) + ρD ′ ( ρ ) / 2 v ′ ( ρ ) < 0 D ′ ( ρ ) < 0 Flat profile ρ ( x ) = ρ 0 unstable if D eff ( ρ 0 ) < 0 Example: v ( ρ ) = v 0 exp( − ρ/ ¯ ρ ) Instability for ρ > ¯ ρ Phase separation of the bacterial colony J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 15 / 27

Surface tension Gradient expansion of ρ ( x ) Phase separation large gradients at the interfaces J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 16 / 27

Surface tension Gradient expansion of ρ ( x ) Phase separation large gradients at the interfaces Expand to higher orders: surface tension − κ ( ρ )∆ 2 ρ ρ = ∇ [ D eff ( ρ ) ∇ ρ ] − κ ∆ 2 ρ ˙ J. Tailleur (CNRS-Univ Paris Diderot) Pattern formation LAFNES 2011 16 / 27